“In our acquisition of knowledge of the Universe (whether mathematical or otherwise) that which renovates the quest is nothing more nor less than complete innocence. It is in this state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.

This unique power is in no way a privilege given to “exceptional talents” – persons of incredible brain power (for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so “endowed at birth, far beyond the ordinary”.

Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the “invisible yet formidable boundaries” that encircle our universe. Only innocence can surmount them, which mere knowledge doesn’t even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child’s play.”

Here is an extract from the blog of another Fields Medallist, Tim Gowers. These comments are perhaps illustrative of the difficulties faced by mathematicians and give others not similarly endowed a great measure of confidence and boost in their studies.

“This raises an important point. I think what may make some mathematicians learn in a different way from others is that some people find straight memorization easier than others. I myself find it difficult, so I don’t really learn anything properly unless I’ve gone through a sort of personal process of rediscovery. That takes time, and the result for me was that although I did adequately as an undergraduate, I was by no means the top in my year at Cambridge — if you’d like to know, I was 15th in my finals — and after the exams I forgot a lot of what I had crammed into my head. I was drawn to the areas of Banach spaces and combinatorics for two reasons. First, and more obviously, I was brilliantly taught in those areas by Béla Bollobás. Secondly, in both areas there were many interesting problems that one could realistically tackle without having to learn a lot of machinery first. Such success as I have had in those areas is no evidence at all for any ability in learning mathematics, where I think I am pretty average: I’m sometimes quick, especially when I’ve thought along similar lines already, but if the area is completely unfamiliar then I’m not quick at all (relative to other mathematicians).

My undergraduate days left me afraid of many subjects: complex analysis, measure theory, most of algebra and almost all geometry, for example. Little by little I have lost quite a lot of that fear: I was forced to come to terms with complex analysis when I had to teach it, and the same happened with measure theory and some of the more elementary parts of algebra and geometry. Editing the Princeton Companion has helped me a lot too: although I don’t understand all those scary areas like topology, PDEs, Riemannian geometry, and so on, I now know just enough about them to see why they are interesting and important. I think I finally got to grips with the concept of cohomology (of the most elementary kind, I hasten to add) a couple of months ago.

Of course, this is all a side issue really. As it happens, I am quite a good example of the kind of person who is greatly helped by having examples first in order to understand generalities. But my case doesn’t rely on that, so if you don’t believe it then take a look at some of the comments, which suggest that there is a significant percentage of mathematicians who feel the same way, and also take a look at my explanation of why I think it is helpful to have examples first.”

These are the words of Gowers himself.

His web page has a lot of interesting discussions about various aspects of elementary mathematics.

M & P:How do you prefer to work on mathematical
problems? Alone or in collaboration?
Okounkov: Perhaps you can guess from what I
said before that I like to work alone, I equally like to
freely share my thoughts, and I also like to perfect
my papers and talks.
There may well be alternate routes, but I personally don’t know how one
can understand something
without both thinking about it quietly over and over
and discussing it with friends.When I feel puzzled,
I like long walks or bike rides. I like to be alone with
my computer playing with formulas or experimenting with
code. But when I finally have an idea, I can’t
wait to share it with others. I am so fortunate to be
able to share my work and my excitement about it
with many brilliant people who are at the same time
wonderful friends.

M & P: Do you have any other interests besides
mathematics?
Werner: I often go to concerts (classical music)
and play(at a nonprofessional level, though) the violin.
Often, I hear people saying”yes,math and music
are so similar, that is why so many mathematicians
are also musicians”. I think that this is only partially
true. I cannot forget that many of those I was playing
music with as a child simply had to stop playing
as adults because their profession did not leave
any time or energy to continue to practice their instruments:
doctors usually have many more working
hours than we do. Also,music is nicely compatible
with mathematics because—at least for me—it
is hard to concentrateon a math problem more than
4-5hours a day,and music is a good complementary
activity: it does not fill the brain with other concerns
and problems that distract from math. It is hard to
do math after having had an argument with somebody
about non-mathematical things, but after one
hour of violin scales,one is in a good state of mind.
Also, but this is a more personal feeling,with the
years, I guess that what I am looking for in music
becomes less and less abstract and analytical and
more and more about emotions—which makes it
less mathematical. . .
But I should also mention that, as far as I can see
it, mathematics is simultaneously an abstract theory
and also very human: When we work on mathematical
ideas, we do it because in a way, we like
them, because we find something in them that resonates
in us (for different reasons, we are all different).
It is not a dry subject that is separate from
the emotional world. This is not so easy to explain
to non mathematicians, for whom this field is just
about computing numbers and solving equations.

This is a classic answer given by Terry Tao in a different interview available here

What is happiness to you—and have you found it?
A: Tolstoy once said that happy families are all alike, but
each unhappy family is unhappy in its own way. I think
the most lasting type of happiness is not the one based
on any sort of achievement, activity, or relationship,
but simply the more mundane type of happiness that
comes from contentment—the absence of stress, discord,
misery, need, self-doubt, bitterness, anger, or
other sources of unhappiness. Of course, if you do take
pleasure in some achievement or relationship, then so
much the better, but it should not define your happiness
to the extent that any hitch in that achievement or
relationship causes you undue grief. I’m quite content
with my own life, and also have the luck to enjoy my
work, my family, and the company of my friends, so I
would consider myself very happy.

I find these particularly illuminating. Also please note that the links to the papers (which have appeared in respective journals) require subscription from some kind and may work only from some university computers or public libraries etc;